50 Dr. J. N. Kruseman on the Potential of the 



Construct a spherical surface through one of the given 

 points and the rim of the bowl. A part of this surface and 

 the bowl determine a limited space, which does not contain 

 the chosen point ; when the other point lies within that space 

 the angle in the first term is obtuse ; when its image is within 

 it the second angle is obtuse ; in all other cases the angles are 

 acute (compare fig. 4). 



§ 9. For the particular case that the bowl becomes a plane 

 disk, W the image of B is with B on the same perpendicular 

 and at the same distance from the plane of the disk. More- 

 over 



m = l, « = 0. 



Thus 

 V 



- E E ! 1 tan~^ ±C ' U I itan-i ±C '^ V (12) 



This is the solution of the problem that Sir W. Thomson 

 had in view when he said, at the end of the cited memoir, 

 " It would be interesting to continue the analytical investi- 

 gation far enough to determine the electric potential at any 

 point in the neighbourhood of a disk electrified under 

 influence." 



The same formula holds for the case of an infinite plane 

 with a circular aperture. The difference lies in the space 

 above defined, as is shown in figs. 5 and 6. 



Formula (11) may, of course, be applied to the case of a 

 spherical surface without any aperture, and the known formulas 

 are then easily found. 



§ 10. Perhaps it will be not uninteresting to give the ex- 

 pansions of formula (2) in spherical harmonics. I shall not, 

 however, show how to get the series ; but I shall write the 

 equation down, and afterwards demonstrate its truth. It takes 

 the following forms : — 



for r<a, V'= ^T^ Y{ ^ + ™(» + V" U.0); 



•tt ,i =0 \ a J A n n + 1 ) T 



for r>a, V= ^T^fi ^ + £^±21^ U {0 ) ; 

 7T „ =0 \r / I n n + \ J T 



K13) 



where <j> n is a zonal harmonic, the coefficient of x n in the 

 expansion of (1 + ^ 2 — 2% cos 6) " 5 . 



I shall presently show that the function defined by equa- 

 tions (13) has the following properties: — 



